Second Derivative

The derivative of a function is also a function. If f(x) is differentiable then f'(x) is also a function of x. If f'(x) is also differentiable, then we can find its derivative function as well which is denoted by f"(x). Thus the process of differentiation can be continued successively as long as each derivative in the sequence is differentiable. The higher order derivatives so found by repeated differentiation play an important role in the study and application of Calculus

The second derivative is the derivative function of the first derivative and it is denoted by f"(x). Second derivative is used determining the Extreme values of the functions and its concavity.In general, the second derivative can be interpreted as the rate of change
of rate of change. If s (t) the position of a moving particle a
function of t, then the velocity v(t) is given by the first derivative
s'(t) and the acceleration a(t) by the second derivative s"(t).

The process of finding the second derivative is rather simple and consists of only two steps.

Find the derivative function f'(x)

Differentiate f'(x) to get the second derivative f"(x).

Example:Find the second derivative of f(x) = 2x3 - 3x2 + 4x +5Step 1: Find the first derivative f'(x) Differentiating the given polynomial function using power rule f'(x) = 6x2 - 6x + 4Step 2: Differentiate f'(x) to get the second derivative We again use the power formula to differentiate f'(x). The second derivative f"(x) = 12x - 6.

When the graph of the second derivative of a function is analyzed some information about the function itself are got.

A function f(x) is concave up wherever the graph of the second derivative is above the x axis.

f(x) is concave down where ever the graph of the second derivative is below the x axis.

The points where the graph of the second derivative crosses the x axis are possible inflection points.

The Inflection points may also occur where the graph of the second derivative display infinite jump, from positive to negative or vice-versa.

let us analyze two Second derivative graphs and infer the behavior of the original function.

In the adjoining graph f"(x) is positive in the intervals (-1,0) and (1, ∞) and negative in the intervals (-∞, -1) and (0, 1).Hence the function f(x) is concave up in (-1,0) and (1, ∞) and concave down in ( -∞, -1) and (-1,0). f(x) and also has possible points of inflection at x = -1, 0 and 1.

In the adjoining graph of second derivative we observe thatf"(x) alternate signs between successive intervals oflength π. Positive in (0, π) and negative in (-π, 0) and (π, 2π).Hence f(x) is concave up in (0, π) and concave down in(-π,0) and (π, 2π). The points where f"(x) is undefined and result in infinitejumps of the graph from top to down and vice versa like x = -π, 0, π and 2π are possible inflection points

Thus if the acceleration function is given, then velocity and position functions can also be known roughly.

Functions are often defined using parameters, meaning both the variables x and y are defined as functions of a third variable say t.

If x = g(t) and y = f(t) the first derivative $\frac{dy}{dx}$ is given by the rule,$\frac{dy}{dx}$ = $\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ = $\frac{f'(t)}{g'(t)}$The second derivative $\frac{d^{2}y}{dx^{2}}$ is got by using the formula,